Let $R$ and $S$ be rings, $M$ be an $R$-module, $N$ be an $S$-module, and $\varphi:R\sqcup M\rightarrow S\sqcup N$ be a map satisfying:
- $\varphi\restriction_R:R\rightarrow S$ is a ring homomorphism
- $\varphi\restriction_M:M\rightarrow N$ is a group homomorphism
- $\varphi(am)=\varphi(a)\varphi(m)$ for all $a\in R, m\in M$
In the special case where $R=S$ and $\varphi\restriction_R$ is the identity map on R, we call $\varphi\restriction_M$ an $R$-module homomorphism. Is there something I am missing that makes $R$-module homomorphisms the only interesting example of such maps, or is this a worthy generalization? Could we define the category Mod of modules using these maps as the morphisms?